Interaction of Aluminum(III) with water. An ab initio study

( )Interaction of Aluminum III with Water.An Ab Initio Study

ALBERT BAKKER,1 KERSTI HERMANSSON,1 JAN LINDGREN,1

MICHAEL M. PROBST,2 PHILIPPE A. BOPP 3

1 ˚Inorganic Chemistry, The Angstrom Laboratory, Uppsala University, Box 538, S-751 21 Uppsala,¨Sweden2 Institute of General, Inorganic and Theoretical Chemistry, Innsbruck University, Innrain 52a,A-6020 Innsbruck, Austria3 Laboratoire de Spectroscopie Moleculaire et Cristalline, Universite de Bordeaux I, 351 Cours de la´Liberation, F-33405 Talence Cedex, France´

Received 3 March 1999; accepted 11 March 1999

3q w Ž . x3qABSTRACT: Hydrated Al ions Al H O , n s 1]6, were examined with ab2 nŽ .initio self-consistent field SCF calculations. The relative contributions of two-, three-,

w Ž . x3qand higher-body terms to the total interaction energy for an Al H O complex were2 6calculated. The sum of all three-body contributions amounts to ; 30% of the sum of allpair-additive contributions and is opposite in sign. The three-body energy contributions

w Ž . x3qwere also derived for two types of Al H O complexes. In the first type, both water2 2molecules reside in the first hydration shell of Al3q and in the second type there is one inthe first shell and one in the second. Altogether 15,500 triplets were investigated andanalytical two- and three-body potential energy functions were derived via a fittingprocedure. Q 1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 659]669, 1999

Key words: ion-water interaction; intermolecular potential functions; nonadditivity; abŽ .initio calculations; aluminium III

Correspondence to: K. Hermansson and M. M. Probst.Contract grant sponsor: SCIENCE ‘‘TWINNING.’’Contract grant number: ERBSCI*CTO000567.Contract grant sponsor: Austrian FWF, Swedish NFR.Contract grant number: P9010-MOB.

( )International Journal of Quantum Chemistry, Vol. 75, 659]669 1999Q 1999 John Wiley & Sons, Inc. CCC 0020-7608 / 99 / 040659-11

BAKKER ET AL.

Introduction

he interaction of metal ions with water is ofT interest because important chemical and bio-physical phenomena are directly related to thebehavior of electrolyte solutions. Moreover, knowl-edge of the ion]water potential energy surface is aprerequisite for the use of computer simulationmethods, which then give new insight into thestructure and dynamics of water molecules around

Ž .ions. Most Monte Carlo MC and molecular dy-Ž .namics MD simulations of ion]water systems

evaluate interaction energies in a pairwise additiveŽ .way, ignoring the higher order ‘‘nonadditive’’

parts of the interaction energy. The most extensiveMD and MC investigations which go beyond thepair model have been performed for liquid water,where the many-body part of the interaction en-ergy is generally relatively modest, as has beenshown by ab initio calculations for water clustersw x1, 2 . Three- and lower-body potentials for water,based on ab initio calculations of triplets of water

w xhave been used in computer simulations 3]6 . Ithas been found that the static properties of water,such as the structure as demonstrated by the dis-tribution functions, do not change as much as thedynamic properties when three-body potentials areapplied.

One can expect the nonadditivity to be moreimportant in ionic solutions than in pure water.

w xClementi et al. 7 investigated the three-body in-qŽ .teraction in Li H O and constructed a potential2 2

with three-body interaction terms; the three-bodyinteractions decreased the binding energy of

qŽ .Li H O , n s 4]6, clusters by ; 15%. Curtiss2 nw xet al. 8, 9 found a too high coordination number

of water around Fe2q and Fe3q due to the neglectof many-body terms. Similar conclusions weredrawn by Probst et al. in a study of the hydration

2q w xof the Be ion 10, 11 . Without three-body inter-actions included, they found a hydration numberof 6, which decreased to 4 when three-body forces

w xwere included 12, 13 . The experimental valueis 4.

Few studies of the importance of ion]waterinteractions beyond the three-body term have been

w xmade for hexacoordinated cations. Probst et al. 14studied the interactions of Naq, Mg2q, and Al3q

with water and showed that the four-body term isquite small, especially for the mono- and divalent

cations; however, this was a limited study. Koll-man and Kuntz drew a similar conclusion for Be2q

w xinteractions in water 15 .The importance of many-body effects in aque-

ous solutions of the trivalent Al ion is expected tobe even more severe than in the cases discussedabove. With the final aim of performing MD simu-lations of hydration structure and OH vibrationalspectra in such solutions, we have, in the presentwork, used ab initio methods to evaluate the mag-nitude of interactions which go beyond the pairapproximation and evaluated whether inclusion ofthree-body potentials appears to be a useful av-enue for the MD simulations. We thus optimized

w Ž . x3qthe structure of the Al H O complex and2 6carried out a systematic study of the two-, three-,four-, five-, and six-body interaction terms. Basedon our findings, we then constructed two- andthree-body analytical potentials for aluminum]water interactions. Ab initio calculations were car-

Ž .ried out at the Hartree]Fock HF level for aboutw Ž . x3q Ž15,500 different orientations of Al H O and2 2

the corresponding Al3q]H O and H O]H O pairs2 2 2.contained within the triplet . Different basis sets

were investigated, as discussed later. A modifiedw xHuzinaga 16 basis set was used for Al, viz. a

w x Ž .3s2 p1d contraction of a 7s4 p1d primitive set,and for O and H Dunning’s double-zeta valence

w xbases 17 were found to be the most appropriateŽ .ones Table II . All interaction energies were cor-

rected for the basis set superposition error on thetwo- and three-body interaction energy by means

w xof the counterpoise method 18 . The ab initiocomputed three-body interaction energies werethen fitted to an analytical expression, producingthe final three-body potential to be used in subse-quent MD simulations. A two-body Al3q]H O2pair potential was also constructed by fitting to

Ž; 800 ab-initio-computed pair energies as de-.scribed below . This two-body potential has subse-

quently been used in MD simulations using alsopair potentials for the other intermolecular inter-actions.

w x 3qIn a recent work 19 , the system Al rH O2was also investigated extensively by means ofquantum chemical calculations. In the course of

3qŽ .developing the Al H O ]H O interaction po-2 6 2tential, the authors performed HF and second-order

Ž . w x3qMøller]Plesset MP2 calculations on Al]H O23qŽ .and Al H O with the aug-cc-pVDZ and, in2 6

some cases, aug-cc-pVQZ basis sets which are suit-w xable for including core]valence correlation 20 .

VOL. 75, NO. 4 / 5660

( )INTERACTION OF Al III WITH WATER

For these mono- and hexa-aqua clusters, our bind-ing energies are overemphasized by about 8% incomparison to the most accurate calculations inw x19 . Additionally, the authors performed a MD

3qŽ .simulation in which the Al H O cluster was2 6treated as the cationic entity. This was done sinceAl3qq H O is not the dissociation product with2the lowest energy at larger distances, and thereforea standard calculation of an analytical Al3qy H O2interaction potential is not possible. The same factcaused us to prevent unwanted electron transfer toAl3q by removing some functions from its basis

Ž .set see below .

Method

According to Hankins, Moskowitz, and Still-w xinger 21]23 the total energy for a system of n

particles can be expanded in series of one-, two-,n-body energies according to:

Ž .E p , p , p , . . . , p1 2 3 n

Ž1. Ž . Ž2. Ž .s E p q E p , pÝ Ýi i ji i-j

Ž3. Ž .q E p , p , p q ???Ý i j ki-j-k

Žn. Ž . Ž .q E p , p , p , . . . , p , 1Ý i j k ni-j-k- . . . -n

where

Ž2. Ž1. Ž1.Ž . Ž . Ž . Ž .E p , p s E p , p y E p q E pi j i j i j

Ž .2

and

Ž3. Ž .E p , p , pi j k

Ž .s E p , p , pi j k

Ž1. Ž1. Ž1.Ž . Ž . Ž .y E p q E p q E pi j k

Ž2. Ž2. Ž2.Ž . Ž . Ž .y E p , p qE p , p qE p , p .i j i k j k

Ž .3

The particles of the interacting system are denotedp , p , p , . . . p ? EŽ1. denotes the energy of an indi-1 2 3 nvidual particle, EŽ2. the two-body interaction ener-gies, and EŽn. the n-body interaction energies. In

w Ž . x3qthe Al H O cluster the seven interacting par-2 6ticles are the Al3q ion and the six H O molecules.2

Most of the ab initio calculations were donew xusing the HONDO-8 24 program. The geometry

w Ž . x3qoptimization of the Al H O cluster was done2 6w xwith the GAUSSIAN system of programs 25 . The

basis sets used are discussed below.

Results and Discussion

Al3+]H O TWO-BODY INTERACTION2

We have calculated the interaction energy of anAl3q]H O complex for different Al]O distances2with a fixed internal water geometry. The experi-

˚mental O]H bond length of 0.9572 A and theexperimental bond angle of 104.528 were used.Over 810 conformations with Al]O distances be-

˚tween 1.0 and 9.5 A were calculated, half of theconfigurations being in the distance range between

˚1.5 and 3.0 A. For each distance, different combina-tions of the two polar angles which further definethe position of Al3q with respect to the watermolecule, were used.

Ž .3qFigure 1 shows the Al]H O potential curve2as a function of the Al]O distance in the C2 vcomplex where oxygen points in the direction of

[ ]3+FIGURE 1. SCF results for the Al ]H O complex2( )curves denoted a using different basis sets. The curves

( )denoted 1 and 2 bases i and e, respectively, in Table Iallow for charge transfer whereas the curves denoted 3( )potential g in Table I do not. The curves denoted b

[ ( ) ]3+show the three-body interactions for Al ] H O2 2complexes as described in the text.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 661

BAKKER ET AL.

Al3q. The basis set superposition was corrected bythe counterpoise method. We start by consideringcurve number 1a of Figure 1. Here the standardbasis set 3-21G was used for aluminum and Dun-

w xning’s double-zeta valence basis set 17 for O andŽ .H basis set i in Table I . The net Mulliken charge

˚of Al is q2.5 at an Al]O distance of 1.8 A and˚q2.1 at a distance of 10 A. The influence of the

charge-transfer effect is even more dramatic atŽ 3q 2qlarger Al]O distances Al q H O ª Al q2

q.H O . The physical reason is that for a suffi-2ciently flexible basis set, there is a competitionbetween two or more dissociation limits, with

3qŽ1 . Ž1 .Al S q H O A predominating for short2 12qŽ2 . qŽ 2 .Al]O distances and Al S q H O A pre-2 1

dominating for large Al]O distances. The same isw xtrue for curve 2 a in Figure 1, i.e., Huzinaga’s 16

Ž . Ž .MIDI 433r43 basis set with a 10 s7p1d r² : Ž .3s1 p1d contraction cf. Table I , augmented with

Ž .a polarization function j s 2.0 . The effect of thisunphysical dissociation limit on the three-bodyenergies is shown by curves 1b and 2b. Thesethree body interaction energies were calculated byadding a second water molecule at an O]Al]O

˚angle of 908 and at an Al]O distance of 1.8 A, withboth water dipole moments pointing away fromthe aluminum ion. Since the potential curves 1aand 2 a converge to a wrong reference state forlong distances, the three-body interaction is notcorrectly represented: the three-body interactionenergies 1b and 2b increase at larger Al]O dis-tances. We subsequently modified this MIDI basisset for Al by removing the basis functions of the

Ž .outermost shell 3s and 3 p and splitting the 2 s

and 2 p functions, giving a split valence basis setfor Al3q which does not allow the unwanted chargetransfer. The modified basis set was thus aŽ . ² : Ž7s4 p1d r 3s2 p1d contraction see Tables I and.II and the resulting potential curve is shown in

Ž .Figure 1 number 3a . Curve 3b represents thethree-body interaction energy calculated with thesame basis set as for curve 3a. This three-bodyinteraction gives the desired shape, decreasing tozero at large Al]O distances.

Table I shows two-body energy minima for sev-eral different basis sets on Al, O, and H. Only thepotentials of type g and h show no change of state

˚over the calculated distances up to 10 A. The basisset of type g, with two- and three-body interac-

Ž .tions as shown in Figure 1 numbers 3a and 3b ,was used in all subsequent calculations describedin this work. The basis set superposition errorŽ .BSSE amounts to ; 10% of the total interactionenergy at the Al3q]H O minimum with this basis2set. This error is avoided by using the counterpoisemethod. At the same minimum, the MP2 correc-tion is about 2%. Since this value is relativelysmall, we did not perform any MP2 correctionsroutinely.

[ ( ) ]3+Al H O THREE-BODY INTERACTION2 2

In this section we present the results of theHartree]Fock calculations that were performed for

w Ž . x3qthe Al H O system for about 15,500 different2 2triplet configurations. Two types of configurationsof this three-body system were considered. In the

Ž .first type type 1 the Al]O1 and Al]O2 distances

TABLE IBasis sets and energy minima.a

(2)Al O rE at r min Curve inmin˚( ) ( )Basis set Contraction Basis set Contraction kJ / mol A Label Fig. 1

( ) ( ) ² : ( ) ( ) ² :MIDI* 533153 11s8p1d / 3s2p1d MIDI* 73/7 10s7p1d / 2s1p1d y744.8 1.787 a( ) ( ) ² : ( ) ( ) ² :MIDI 533153 11s8p / 3s2p MIDI 73/7 10s7p / 2s1p y698.4 1.796 b( ) ( ) ² : ( ) ² :MIDI* 533153 11s8p1d / 3s2p1d DZP 10s5p1d / 3s1p1d y735.1 1.760 c

( ) ( ) ² : ( ) ² :MIDI 533153 11s8p / 3s2p DZP 10s5p1d / 3s1p1d y708.3 1.757 d( ) ( ) ² : ( ) ² :MIDI* 433143 10s7p1d / 3s2p1d DZP 10s5p1d / 3s1p1d y724.6 1.759 e 2a, 2b

( ) ( ) ² : ( ) ² :MIDI 433143 10s7p / 3s2p DZP 10s5p1d / 3s1p1d y706.8 1.760 f( ) ( ) ² : ( ) ² :Modified MIDI* 421/31 7s4p1d / 3s2p1d DZP 10s5p1d / 3s1p1d y691.0 1.700 g 3a, 3b( ) ( ) ² : ( ) ² :Modified MIDI 421/31 7s4p / 3s2p DZP 10s5p1d / 3s1p1d y670.3 1.707 h

( ) ² : ( ) ² :3 ]21G 3s6sp / 1s3sp DZP 10s5p1d / 3s1p1d y749.0 1.772 i 1a, 1b

a [ ]The original MIDI and MIDI* bases, corresponding to bases sets a ] f here, are those of Huzinaga 16 . The DZP basis for O is[ ]from Dunning 17 . For H, Dunning’s DZP bases was always used. Our ‘‘modified-MIDI’’ basis sets were derived by changing basis

( )sets e and f by removing the basis functions of the outermost shell 3s and 3p and splitting the 2s and 2p functions.

VOL. 75, NO. 4 / 5662


TABLE II[ ]3s2p1d contracted basis set for aluminum.

( )MIDI* 433/43 Basis set in our work

[ ] [ ]j 3s j 3s

2244.7744 0.0187029 2244.7744 0.0187029338.68320 0.1298302 338.68320 0.1298302

76.402946 0.4551734 76.402946 0.455173420.367550 0.5341938 20.367550 0.534193830.587341 y0.0913928 30.587341 y0.0913928

2.9449883 0.5803227 2.9449883 0.58032271.0582075 0.49430561.3075542 y0.1483246 1.0582075 0.49430560.1635624 0.65768800.0596563 0.4407021

[ ] [ ]j 2p j 2p

65.366461 0.0374231 65.366462708 0.037423114.829354 0.2132589 14.829354286 0.2132589

4.2621974 0.5085636 4.262197495 0.50856361.261241555 0.45156510.040019491 0.3491117 1.2612415 0.45156510.296881765 0.24597150.109109089 0.5363164

[ ] [ ]j 1d j 1d

2.000 1.0 2.000 1.0

˚are between 1.7 and 2.5 A, and the angle Ol]Al]O2is more than 458. An arabic number, with or with-out parenthesis, appended to a chemical symbolwill be used here to identify the molecule to whichan atom belongs. This geometry was chosen sincethe preference for octahedral coordination aroundAl3q in aqueous solutions is well known. Thusthis type of configurations corresponds to two wa-ter molecules residing in the first hydration sphere

3q Ž .of Al . Moreover for each position of H O 2 , this2molecule was allowed to rotate around the threeperpendicular rotation axes in steps of 308. The

Ž .Al]H O 1 part of the triplet was, however, al-2ways kept planar. A representative type 1 configu-ration is shown in Figure 2. The second type ofconfigurations, type 2, corresponds to configura-tions with one water molecule in the first hydra-tion shell and the second in the second hydration

[ ( ) ]3+ ( )FIGURE 2. Schematic drawings of the two types of Al H O configurations see text .2 2


BAKKER ET AL.

shell of Al3q with an O1]Al]O2 angle less than458. In the type 2 configurations the Al]O1 dis-

˚tance was kept fixed at 1.8 A while the O1]O2distance was allowed to vary between 2.0 and 5.3˚ Ž .A. Again, the Al]H O 1 part of the complex was2

Ž .always kept planar, while the H O 2 molecule2

was rotated stepwise. A representative type 2 con-figuration is shown in Figure 2. For type 1 wecalculated the three-body energy for over 4500 andfor type 2 for over 11,000 configurations. The inter-action energies were corrected for the BSSE accord-

Ž .ing to the counterpoise method. Figure 3 a shows

[ ( ) ]3+FIGURE 3. Hartree ]Fock and analytical three-body energy contour plot for planar Al H O type 1 and type 22 2˚( ) ( )complexes. H O 1 is kept fixed at an Al ]O distance of 1.8 A, and the dipole vector of H O 1 is collinear with the2 2

( )vector from Al to O1. For type 1, H O 2 is oriented with its dipole vector pointing in the same direction as the Al ]O22( )vector. For type 2, H O 2 is oriented with its dipole vector collinear with the vector from O1 to O2. The x and y2

coordinates refer to the position of O2. Energies in kJ / mol. Upper plot: the ab initio energies are shown for type 1( 3+) ( 3+) ( )lower left from Al and type 2 upper right from Al . Lower plot: energies from the analytical function b which was

[ ( ) ]only constructed for type 1 Eq. 8 in the text .

VOL. 75, NO. 4 / 5664


the resulting three-body energy surfaces for bothtypes of geometries. This figure, of course, showsonly a section through the multidimensional sur-

Ž .face, viz. the section where H O 1 has a fixed2position and the second water molecule is lying inthe same plane, its dipole moment pointing awayfrom Al3q type 1 or O1 type 2. The figure showsthat the nonadditivity term is not negligible. In

˚fact, when O1]Al]O2 is 908 and r is 1.8 AAl-O2Žwhich corresponds to the energy minimum for the

3q .two-body Al ]H O interaction , the three-body2interaction energy is 17% of the pair-additive in-teraction energy of the triplet. The three-body cor-rections for type 1 are all positive. The positiverepulsive three-body energy is expected from sim-ple electrostatic considerations; the dipoles in-

Ž 3q.duced in the first-shell water molecules from Alinteract ‘‘unfavorably’’ with each other. Type 2configurations show both positive and negativeregions. The negative region is found for the hy-drogen-bonded water molecules. The three-bodyenergies for type 1 configurations depend verylittle on the positions of the hydrogen atoms, incontrast to the type 2 configurations, where theorientation of the molecule changes the three-bodyenergy dramatically, leading to more complicatedthree-body energy surfaces as compared to type 1.

As mentioned, the energies were in all casescorrected for BSSE’s using the counterpoisemethod. The BSSE correction amounts to ; 15%

[ ( ) ]3+FIGURE 4. Al H O cluster at the optimized2 6Hartree ]Fock geometry.

at the Al-O distances where the two-body interac-tion is at a minimum. The MP2 correction, how-

Ž .ever, is small 4% and was therefore not included.

[ ( ) ]3+Al H O CLUSTER2 6

Converted to analytical functions, the ab initiotwo- and three-body interaction energy can beused in computer simulations. Ab initio calcula-tions on Al]water clusters of more than two watermolecules can provide information on the contri-butions from higher order terms to the total inter-action energy. Ab initio calculations of clusters ofvery many water molecules become prohibitivelyCPU-time demanding, but already clusters of alimited size can provide very useful information.We have performed Hartree]Fock calculations onw Ž . x3qAl H O clusters of size up to n s 6. This is2 nthe experimentally determined coordination num-

3q Ž . w xber for Al aq 26]28 . The geometry of thehexahydrated Al3q ion was optimized within theS point group, keeping the O]Al]O angles fixed4at 908 and 1808. Figure 4 shows the optimizedstructure, which has four equatorial water

˚ Žmolecules at an Al]O distance of 1.959 A O1 to. Ž .O4 , and two axial ones O5 and O6 at a distance

˚of 1.957 A; the latter are twisted by 458. The Al]Odistances are considerably longer than in theAl3q]H O complex. This is mainly due to the2repulsion between the water molecules in the

˚hexa-complex. The O]H distances are ; 0.015 Alonger than in a free water molecule with the same

˚Ž .basis set 0.944 A , and the H]O]H angles areabout 18 smaller than in the isolated moleculeŽ .106.88 .

Ž .Figure 5 and Table III first two rows show thetwo- and higher-order interaction energy forw Ž . x3q Ž . Ž .Al H O and H O . For H O , the water2 6 2 6 2 6molecules are frozen in the same positions as

Ž .around the ion. The energies for the H O com-2 6plex are included to single out those interactionswhich do not involve the Al3q ion. We see that the

Ž .interaction energies for the H O cluster are much2 6smaller and of opposite sign compared withw Ž . x3qAl H O . This shows that the overwhelming2 6parts of the two-body, and especially of thehigher-order, interactions originate from interac-tions with the Al3q ion and the mutual polariza-tion of the water molecules is less important. Asthe figure shows, the three-body part of the inter-action energy contributes significantly to the totalinteraction energy of the Al cluster, ; 32%.


BAKKER ET AL.

[ ( ) ]3+ ( ) ( )FIGURE 5. The n-order interaction energies for a Al H O cluster dark bars and a H O cluster with the same2 6 2 6[ ( ) ]3+ ( ) ( ) ( )H O coordinates as the Al H O cluster light bars : a the sum of all interaction energies and b the sum of the2 2 6

( )interaction energies involving a particular water molecule see definitions in Table III .

Higher-body energy terms are comparativelysmall. The same is true for the sum of two- andhigher-body interaction energies involving a singlewater molecule, as defined in the last two rows ofTable III.

ANALYTICAL POTENTIALS

Our procedure for obtaining analytical two- andthree-body potentials may be divided into three

Ž .steps: 1 selection of the conformations and abŽ .initio calculations, 2 selection of the analytical

Ž .form, and 3 the fitting procedure.

Al3+]H O PAIR POTENTIAL2

Many different forms for the Al3q]H O pair2potential were tested. The final expression used isthe following:

Ž2. Ž . Ž .E p , p s V 4Ý Ýi j a b

sitea sitebon particle i on particle j

where

q q Aa b a b Ž . Ž .V s q q B exp yC r . 5a b a b a b a b2r ra b a b

TABLE IIITwo- and higher-body contributions to the total interaction energy and the interaction energy involving one

3+ a( ) [ ( ) ] ( )single water molecule defined on lines 3 and 4 in the table of the Al H O and H O clusters.2 6 2 6

2 3 4 5 6 7

(n) 3 +( ( ) ( ) )E H O 1 , . . . , H O 6 , Al y3896.61 1243.22 y369.33 99.713 y18.95 1.24Ý 2 2all n-mers

(n)( ( ) ( ))E H O 1 , . . . , H O 6 330.75 y60.01 9.21 y0.56 y0.16 }Ý 2 2all n-mers

(n) 3 +( ( ) ( ) )E H O 1 , . . . , H O 6 , Al y649.43 415.24 y181.72 64.74 y15.46 1.24Ý 2 2all n-mers

( )containing H O 12

(n)( ( ) ( ))E H O 1 , . . . , H O 6 114.94 y29.79 6.21 y0.60 y0.16 }Ý 2 2all n-mers

( )containing H O 12

aAll energies in kJ / mol.

VOL. 75, NO. 4 / 5666


Here r is the distance between site a on particlea b

A and site b on particle B; q and q are thea b

charges of the sites and A , B , and C are thea b a b a b

fitted parameters. The charges for O and H werechosen to be y0.65966e and q0.32983e, in accor-dance with the water charges employed in the

Ž .Bopp]Jancso]Heinzinger BJH water]water po-w xtentials 29 .

w xThe fits were done with the program NSGB 30 .The potential functions obtained after the fitting

˚Ž .procedure were in kJrmol and r in A :

Ž . 2V r s y2750.50rr y 2495.69rrAlO

Ž . Ž .q 266000 exp y3.89948rr , 6

Ž . 2V r s 1375.26rr q 160.657rrAlH

Ž . Ž .q 287.456 exp y0.35461rr . 7

The quality of the fit is demonstrated in Figures 6Ž .and 7 a which show that the fit is satisfactory for

all geometries included in the fitting procedure.The largest deviations between self-consistent fieldŽ .SCF and fitted energies occur for the repulsiveconformations. However, these highly repulsiveregions are normally unimportant in a computer

Ž .simulation, since they are seldom never sampled.

[ ( ) ]3+Al H O Three-Body Potential2 2

The three-body potential function was extractedw Ž . x3qfrom the energies of the Al H O triplets of2 6

w xthe type 1 form. The following expression 10, 11

FIGURE 6. Comparison between SCF energy points( ) ( )dots and analytical potential curves lines for threedirections of the dipole moment of water with respect to

3+ ( ) ( )the distance vector from Al to O, 08 O , 908 v , and( )1808 I .

was used:

22Ž3. 2 2Ž Ž . . Ž .E s A B q p y a exp yC r q r ,Ž .AlO1 AlO 2

Ž .8

where r and r are the distances betweenAlO1 AlO2Al3q and the oxygen atoms of the water molecules.The variable a denotes the O1]Al]O2 angle andreflects the fact that the three-body interactionenergy is strongly dependent on this angle. In all

FIGURE 7. Correlation between interaction energies from the SCF calculations and those derived from analytical( ) ( )expressions as described in the text: a two-body and b three-body interactions.


BAKKER ET AL.

configurations used, the dipole moments of thewater molecules point outward from the Al]Odirection, which is energetically the most favorabledirection for the water molecules. In applying thispotential in a molecular dynamics simulation, thisstrong orientational dependence will not occur for

˚Ž .larger Al]O distances ) 3.5 A but, as Figure 3Ž .shows and as the functional form in 8 reflects,

the three-body interaction energy decays very fastwith Al]O distance. The fitting procedure donewith the subroutine NSGB, gave the followingparameters: A s 74.85 kJrmol, B s 0.06413, and

˚y2C s 0.2465 A . The quality of the fit is demon-Ž .strated in Figure 7 b , which shows the correlation

Ž .between SCF and fitted energies. Figure 2 b showsthe fitted function. The standard deviation ob-tained from the fitting procedure was 12 kJrmol;this is good considering that about 4500 pointshave been fitted to a relatively simple functionalform. In contrast to the type 1 configurations, thetype 2 triplets show a stronger dependence on theorientation of the dipole moment of both watermolecules. Expressing three-body energies in ananalytical form thus needs more parameters thanthe ion]water distances and the O]Al]O angleonly.

Conclusions

We have investigated the Al3q]H O and2w Ž . x3qAl H O interaction energies by ab initio SCF2 6

calculations. We constructed an appropriate basisset for Al and O that does not allow for chargetransfer. Test calculations were made without cor-recting for basis set superposition errors and byperforming a counterpoise correction using the fullbasis set for all the calculated di- and trimers. Thebasis set superposition error was found to be non-negligible for our purposes, and all our interactionenergies were corrected with the counterpoisemethod. The effect of the Møller]Plesset second-order perturbation was, however, found to be smallcompared to the basis set superposition error.Based on these investigations, we calculated theeffect of Al3q on the two- and higher-order inter-

w Ž . x3qaction energy terms of an Al H O cluster.2 6

The three-body interactions were found to be onethird the size of the sum of the two-body interac-tions. The four- and higher- body contributions onthe interaction energy are much smaller.

We have presented ion]water and ion]water]water analytical potentials which were derived byfitting to the ab initio interaction energies. Ournew potentials are currently being used in MD

Ž .simulations of AlCl aq solution.3

ACKNOWLEDGMENT

Financially support by a SCIENCE ‘‘TWIN-wNING’’ grant from the EEC grant number ERB-

Ž .xSCI*CTO00567- SC1000567 and the Austrian FWFŽ .project P9010-MOB and the Swedish Natural Sci-

Ž .ence Research Council NFR are gratefully ac-knowledged.

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Interaction of Aluminum(III) with water. An ab initio study